Integrand size = 21, antiderivative size = 616 \[ \int \frac {\sec (c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=-\frac {\sqrt [3]{b} \left (a^{4/3}-2 b^{4/3}\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} \left (a^2-b^2\right ) d}-\frac {\sqrt [3]{b} \left (a^2-2 a^{2/3} b^{4/3}+b^2\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} \left (a^2-b^2\right )^2 d}-\frac {\log (1-\sin (c+d x))}{2 (a+b)^2 d}+\frac {\log (1+\sin (c+d x))}{2 (a-b)^2 d}-\frac {\sqrt [3]{b} \left (a^{4/3}+2 b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} \left (a^2-b^2\right ) d}-\frac {\sqrt [3]{b} \left (a^2+2 a^{2/3} b^{4/3}+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 \sqrt [3]{a} \left (a^2-b^2\right )^2 d}+\frac {\sqrt [3]{b} \left (a^{4/3}+2 b^{4/3}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{18 a^{5/3} \left (a^2-b^2\right ) d}+\frac {\sqrt [3]{b} \left (a^2+2 a^{2/3} b^{4/3}+b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 \sqrt [3]{a} \left (a^2-b^2\right )^2 d}-\frac {2 a b \log \left (a+b \sin ^3(c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}+\frac {b}{3 \left (a^2-b^2\right ) d \left (a+b \sin ^3(c+d x)\right )}-\frac {b \sin (c+d x) (b-a \sin (c+d x))}{3 a \left (a^2-b^2\right ) d \left (a+b \sin ^3(c+d x)\right )} \] Output:
-1/9*b^(1/3)*(a^(4/3)-2*b^(4/3))*arctan(1/3*(a^(1/3)-2*b^(1/3)*sin(d*x+c)) *3^(1/2)/a^(1/3))*3^(1/2)/a^(5/3)/(a^2-b^2)/d-1/3*b^(1/3)*(a^2-2*a^(2/3)*b ^(4/3)+b^2)*arctan(1/3*(a^(1/3)-2*b^(1/3)*sin(d*x+c))*3^(1/2)/a^(1/3))*3^( 1/2)/a^(1/3)/(a^2-b^2)^2/d-1/2*ln(1-sin(d*x+c))/(a+b)^2/d+1/2*ln(1+sin(d*x +c))/(a-b)^2/d-1/9*b^(1/3)*(a^(4/3)+2*b^(4/3))*ln(a^(1/3)+b^(1/3)*sin(d*x+ c))/a^(5/3)/(a^2-b^2)/d-1/3*b^(1/3)*(a^2+2*a^(2/3)*b^(4/3)+b^2)*ln(a^(1/3) +b^(1/3)*sin(d*x+c))/a^(1/3)/(a^2-b^2)^2/d+1/18*b^(1/3)*(a^(4/3)+2*b^(4/3) )*ln(a^(2/3)-a^(1/3)*b^(1/3)*sin(d*x+c)+b^(2/3)*sin(d*x+c)^2)/a^(5/3)/(a^2 -b^2)/d+1/6*b^(1/3)*(a^2+2*a^(2/3)*b^(4/3)+b^2)*ln(a^(2/3)-a^(1/3)*b^(1/3) *sin(d*x+c)+b^(2/3)*sin(d*x+c)^2)/a^(1/3)/(a^2-b^2)^2/d-2/3*a*b*ln(a+b*sin (d*x+c)^3)/(a^2-b^2)^2/d+1/3*b/(a^2-b^2)/d/(a+b*sin(d*x+c)^3)-1/3*b*sin(d* x+c)*(b-a*sin(d*x+c))/a/(a^2-b^2)/d/(a+b*sin(d*x+c)^3)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 5.00 (sec) , antiderivative size = 564, normalized size of antiderivative = 0.92 \[ \int \frac {\sec (c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\frac {\frac {12 \sqrt {3} \sqrt [3]{a} b^{5/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\left (a^2-b^2\right )^2}+\frac {4 \sqrt {3} b^{5/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{5/3} \left (a^2-b^2\right )}-\frac {9 \log (1-\sin (c+d x))}{(a+b)^2}+\frac {9 \log (1+\sin (c+d x))}{(a-b)^2}-\frac {12 \sqrt [3]{a} b^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{\left (a^2-b^2\right )^2}-\frac {4 b^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{a^{5/3} \left (a^2-b^2\right )}+\frac {6 \sqrt [3]{a} b^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{\left (a^2-b^2\right )^2}+\frac {2 b^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{a^{5/3} \left (a^2-b^2\right )}-\frac {12 a b \log \left (a+b \sin ^3(c+d x)\right )}{\left (a^2-b^2\right )^2}+\frac {9 b \left (a^2+b^2\right ) \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},-\frac {b \sin ^3(c+d x)}{a}\right ) \sin ^2(c+d x)}{a \left (a^2-b^2\right )^2}+\frac {9 b \operatorname {Hypergeometric2F1}\left (\frac {2}{3},2,\frac {5}{3},-\frac {b \sin ^3(c+d x)}{a}\right ) \sin ^2(c+d x)}{a^3-a b^2}+\frac {6 b}{\left (a^2-b^2\right ) \left (a+b \sin ^3(c+d x)\right )}-\frac {6 b^2 \sin (c+d x)}{a \left (a^2-b^2\right ) \left (a+b \sin ^3(c+d x)\right )}}{18 d} \] Input:
Integrate[Sec[c + d*x]/(a + b*Sin[c + d*x]^3)^2,x]
Output:
((12*Sqrt[3]*a^(1/3)*b^(5/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sq rt[3]*a^(1/3))])/(a^2 - b^2)^2 + (4*Sqrt[3]*b^(5/3)*ArcTan[(a^(1/3) - 2*b^ (1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))])/(a^(5/3)*(a^2 - b^2)) - (9*Log[1 - Sin[c + d*x]])/(a + b)^2 + (9*Log[1 + Sin[c + d*x]])/(a - b)^2 - (12*a^(1 /3)*b^(5/3)*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(a^2 - b^2)^2 - (4*b^(5/3 )*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(a^(5/3)*(a^2 - b^2)) + (6*a^(1/3)* b^(5/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^ 2])/(a^2 - b^2)^2 + (2*b^(5/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2])/(a^(5/3)*(a^2 - b^2)) - (12*a*b*Log[a + b*Sin[c + d*x]^3])/(a^2 - b^2)^2 + (9*b*(a^2 + b^2)*Hypergeometric2F1[2/3, 1, 5/3 , -((b*Sin[c + d*x]^3)/a)]*Sin[c + d*x]^2)/(a*(a^2 - b^2)^2) + (9*b*Hyperg eometric2F1[2/3, 2, 5/3, -((b*Sin[c + d*x]^3)/a)]*Sin[c + d*x]^2)/(a^3 - a *b^2) + (6*b)/((a^2 - b^2)*(a + b*Sin[c + d*x]^3)) - (6*b^2*Sin[c + d*x])/ (a*(a^2 - b^2)*(a + b*Sin[c + d*x]^3)))/(18*d)
Time = 0.94 (sec) , antiderivative size = 561, normalized size of antiderivative = 0.91, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3702, 7276, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sec (c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos (c+d x) \left (a+b \sin (c+d x)^3\right )^2}dx\) |
| \(\Big \downarrow \) 3702 |
| \(\displaystyle \frac {\int \frac {1}{\left (1-\sin ^2(c+d x)\right ) \left (b \sin ^3(c+d x)+a\right )^2}d\sin (c+d x)}{d}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {\int \left (\frac {b \left (b \sin ^2(c+d x)-a \sin (c+d x)+b\right )}{\left (b^2-a^2\right ) \left (b \sin ^3(c+d x)+a\right )^2}-\frac {1}{2 (a+b)^2 (\sin (c+d x)-1)}+\frac {1}{2 (a-b)^2 (\sin (c+d x)+1)}+\frac {b \left (-2 a b \sin ^2(c+d x)+\left (a^2+b^2\right ) \sin (c+d x)-2 a b\right )}{\left (a^2-b^2\right )^2 \left (b \sin ^3(c+d x)+a\right )}\right )d\sin (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {b (a-\sin (c+d x) (b-a \sin (c+d x)))}{3 a \left (a^2-b^2\right ) \left (a+b \sin ^3(c+d x)\right )}-\frac {2 a b \log \left (a+b \sin ^3(c+d x)\right )}{3 \left (a^2-b^2\right )^2}-\frac {\sqrt [3]{b} \left (-2 a^{2/3} b^{4/3}+a^2+b^2\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} \left (a^2-b^2\right )^2}-\frac {\sqrt [3]{b} \left (a^{4/3}-2 b^{4/3}\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} \left (a^2-b^2\right )}+\frac {\sqrt [3]{b} \left (2 a^{2/3} b^{4/3}+a^2+b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 \sqrt [3]{a} \left (a^2-b^2\right )^2}+\frac {\sqrt [3]{b} \left (a^{4/3}+2 b^{4/3}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{18 a^{5/3} \left (a^2-b^2\right )}-\frac {\sqrt [3]{b} \left (2 a^{2/3} b^{4/3}+a^2+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 \sqrt [3]{a} \left (a^2-b^2\right )^2}-\frac {\sqrt [3]{b} \left (a^{4/3}+2 b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} \left (a^2-b^2\right )}-\frac {\log (1-\sin (c+d x))}{2 (a+b)^2}+\frac {\log (\sin (c+d x)+1)}{2 (a-b)^2}}{d}\) |
Input:
Int[Sec[c + d*x]/(a + b*Sin[c + d*x]^3)^2,x]
Output:
(-1/3*(b^(1/3)*(a^(4/3) - 2*b^(4/3))*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d *x])/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(5/3)*(a^2 - b^2)) - (b^(1/3)*(a^2 - 2 *a^(2/3)*b^(4/3) + b^2)*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3] *a^(1/3))])/(Sqrt[3]*a^(1/3)*(a^2 - b^2)^2) - Log[1 - Sin[c + d*x]]/(2*(a + b)^2) + Log[1 + Sin[c + d*x]]/(2*(a - b)^2) - (b^(1/3)*(a^(4/3) + 2*b^(4 /3))*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(9*a^(5/3)*(a^2 - b^2)) - (b^(1/ 3)*(a^2 + 2*a^(2/3)*b^(4/3) + b^2)*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(3 *a^(1/3)*(a^2 - b^2)^2) + (b^(1/3)*(a^(4/3) + 2*b^(4/3))*Log[a^(2/3) - a^( 1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2])/(18*a^(5/3)*(a^2 - b^ 2)) + (b^(1/3)*(a^2 + 2*a^(2/3)*b^(4/3) + b^2)*Log[a^(2/3) - a^(1/3)*b^(1/ 3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2])/(6*a^(1/3)*(a^2 - b^2)^2) - (2* a*b*Log[a + b*Sin[c + d*x]^3])/(3*(a^2 - b^2)^2) + (b*(a - Sin[c + d*x]*(b - a*Sin[c + d*x])))/(3*a*(a^2 - b^2)*(a + b*Sin[c + d*x]^3)))/d
Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x _)])^(n_))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Si mp[ff/f Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*(c*ff*x)^n)^p, x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[(m - 1)/2] && (EqQ[n, 4] || GtQ[m, 0] || IGtQ[p, 0] || IntegersQ[m, p])
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 2.51 (sec) , antiderivative size = 389, normalized size of antiderivative = 0.63
| method | result | size |
| derivativedivides | \(\frac {\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{2 \left (a -b \right )^{2}}-\frac {\ln \left (\sin \left (d x +c \right )-1\right )}{2 \left (a +b \right )^{2}}+\frac {b \left (\frac {\left (\frac {a^{2}}{3}-\frac {b^{2}}{3}\right ) \sin \left (d x +c \right )^{2}-\frac {b \left (a^{2}-b^{2}\right ) \sin \left (d x +c \right )}{3 a}+\frac {a^{2}}{3}-\frac {b^{2}}{3}}{a +b \sin \left (d x +c \right )^{3}}+\frac {\frac {2 \left (-4 a^{2} b +b^{3}\right ) \left (\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin \left (d x +c \right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3}+\frac {2 \left (2 a^{3}+b^{2} a \right ) \left (-\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\sin \left (d x +c \right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3}-\frac {2 a^{2} \ln \left (a +b \sin \left (d x +c \right )^{3}\right )}{3}}{a}\right )}{\left (a -b \right )^{2} \left (a +b \right )^{2}}}{d}\) | \(389\) |
| default | \(\frac {\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{2 \left (a -b \right )^{2}}-\frac {\ln \left (\sin \left (d x +c \right )-1\right )}{2 \left (a +b \right )^{2}}+\frac {b \left (\frac {\left (\frac {a^{2}}{3}-\frac {b^{2}}{3}\right ) \sin \left (d x +c \right )^{2}-\frac {b \left (a^{2}-b^{2}\right ) \sin \left (d x +c \right )}{3 a}+\frac {a^{2}}{3}-\frac {b^{2}}{3}}{a +b \sin \left (d x +c \right )^{3}}+\frac {\frac {2 \left (-4 a^{2} b +b^{3}\right ) \left (\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin \left (d x +c \right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3}+\frac {2 \left (2 a^{3}+b^{2} a \right ) \left (-\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\sin \left (d x +c \right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3}-\frac {2 a^{2} \ln \left (a +b \sin \left (d x +c \right )^{3}\right )}{3}}{a}\right )}{\left (a -b \right )^{2} \left (a +b \right )^{2}}}{d}\) | \(389\) |
| risch | \(-\frac {i x}{a^{2}-2 a b +b^{2}}-\frac {i c}{d \left (a^{2}-2 a b +b^{2}\right )}+\frac {i x}{a^{2}+2 a b +b^{2}}+\frac {i c}{d \left (a^{2}+2 a b +b^{2}\right )}+\frac {4 i a^{6} b \,d^{3} x}{a^{9} d^{3}-2 b^{2} a^{7} d^{3}+a^{5} b^{4} d^{3}}+\frac {4 i a^{6} b \,d^{2} c}{a^{9} d^{3}-2 b^{2} a^{7} d^{3}+a^{5} b^{4} d^{3}}-\frac {2 b \left (i a \,{\mathrm e}^{5 i \left (d x +c \right )}-6 i a \,{\mathrm e}^{3 i \left (d x +c \right )}+2 b \,{\mathrm e}^{4 i \left (d x +c \right )}+i a \,{\mathrm e}^{i \left (d x +c \right )}-2 b \,{\mathrm e}^{2 i \left (d x +c \right )}\right )}{3 a \left (-a^{2}+b^{2}\right ) d \left ({\mathrm e}^{6 i \left (d x +c \right )} b -3 b \,{\mathrm e}^{4 i \left (d x +c \right )}+3 b \,{\mathrm e}^{2 i \left (d x +c \right )}-8 i a \,{\mathrm e}^{3 i \left (d x +c \right )}-b \right )}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d \left (a^{2}-2 a b +b^{2}\right )}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d \left (a^{2}+2 a b +b^{2}\right )}+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (729 a^{9} d^{3}-1458 b^{2} a^{7} d^{3}+729 a^{5} b^{4} d^{3}\right ) \textit {\_Z}^{3}+729 a^{6} b \,d^{2} \textit {\_Z}^{2}+27 a^{3} b^{2} d \textit {\_Z} +8 a^{2} b -b^{3}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\left (\frac {324 i a^{11} d^{2}}{8 a^{6} b +28 a^{4} b^{3}-10 a^{2} b^{5}+b^{7}}-\frac {486 i a^{9} b^{2} d^{2}}{8 a^{6} b +28 a^{4} b^{3}-10 a^{2} b^{5}+b^{7}}+\frac {162 i a^{5} b^{6} d^{2}}{8 a^{6} b +28 a^{4} b^{3}-10 a^{2} b^{5}+b^{7}}\right ) \textit {\_R}^{2}+\left (\frac {216 i a^{8} b d}{8 a^{6} b +28 a^{4} b^{3}-10 a^{2} b^{5}+b^{7}}+\frac {396 i a^{6} b^{3} d}{8 a^{6} b +28 a^{4} b^{3}-10 a^{2} b^{5}+b^{7}}-\frac {144 i a^{4} b^{5} d}{8 a^{6} b +28 a^{4} b^{3}-10 a^{2} b^{5}+b^{7}}+\frac {18 i a^{2} b^{7} d}{8 a^{6} b +28 a^{4} b^{3}-10 a^{2} b^{5}+b^{7}}\right ) \textit {\_R} -\frac {28 i a^{5} b^{2}}{8 a^{6} b +28 a^{4} b^{3}-10 a^{2} b^{5}+b^{7}}+\frac {10 i a^{3} b^{4}}{8 a^{6} b +28 a^{4} b^{3}-10 a^{2} b^{5}+b^{7}}\right ) {\mathrm e}^{i \left (d x +c \right )}-\frac {8 a^{6} b}{8 a^{6} b +28 a^{4} b^{3}-10 a^{2} b^{5}+b^{7}}-\frac {28 a^{4} b^{3}}{8 a^{6} b +28 a^{4} b^{3}-10 a^{2} b^{5}+b^{7}}+\frac {10 a^{2} b^{5}}{8 a^{6} b +28 a^{4} b^{3}-10 a^{2} b^{5}+b^{7}}-\frac {b^{7}}{8 a^{6} b +28 a^{4} b^{3}-10 a^{2} b^{5}+b^{7}}\right )\right )\) | \(946\) |
Input:
int(sec(d*x+c)/(a+b*sin(d*x+c)^3)^2,x,method=_RETURNVERBOSE)
Output:
1/d*(1/2/(a-b)^2*ln(1+sin(d*x+c))-1/2/(a+b)^2*ln(sin(d*x+c)-1)+b/(a-b)^2/( a+b)^2*(((1/3*a^2-1/3*b^2)*sin(d*x+c)^2-1/3*b*(a^2-b^2)/a*sin(d*x+c)+1/3*a ^2-1/3*b^2)/(a+b*sin(d*x+c)^3)+2/3/a*((-4*a^2*b+b^3)*(1/3/b/(1/b*a)^(2/3)* ln(sin(d*x+c)+(1/b*a)^(1/3))-1/6/b/(1/b*a)^(2/3)*ln(sin(d*x+c)^2-(1/b*a)^( 1/3)*sin(d*x+c)+(1/b*a)^(2/3))+1/3/b/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1 /2)*(2/(1/b*a)^(1/3)*sin(d*x+c)-1)))+(2*a^3+a*b^2)*(-1/3/b/(1/b*a)^(1/3)*l n(sin(d*x+c)+(1/b*a)^(1/3))+1/6/b/(1/b*a)^(1/3)*ln(sin(d*x+c)^2-(1/b*a)^(1 /3)*sin(d*x+c)+(1/b*a)^(2/3))+1/3*3^(1/2)/b/(1/b*a)^(1/3)*arctan(1/3*3^(1/ 2)*(2/(1/b*a)^(1/3)*sin(d*x+c)-1)))-a^2*ln(a+b*sin(d*x+c)^3))))
Result contains complex when optimal does not.
Time = 2.23 (sec) , antiderivative size = 10855, normalized size of antiderivative = 17.62 \[ \int \frac {\sec (c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(sec(d*x+c)/(a+b*sin(d*x+c)^3)^2,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {\sec (c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Timed out} \] Input:
integrate(sec(d*x+c)/(a+b*sin(d*x+c)**3)**2,x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 483, normalized size of antiderivative = 0.78 \[ \int \frac {\sec (c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx =\text {Too large to display} \] Input:
integrate(sec(d*x+c)/(a+b*sin(d*x+c)^3)^2,x, algorithm="maxima")
Output:
1/18*(4*sqrt(3)*(2*a^3*((a/b)^(2/3) + 1) - 2*a^2*b*(2*(a/b)^(1/3) + a/b) + a*b^2*(a/b)^(2/3) + b^3*(a/b)^(1/3))*arctan(-1/3*sqrt(3)*((a/b)^(1/3) - 2 *sin(d*x + c))/(a/b)^(1/3))/((a^5*(a/b)^(2/3) - 2*a^3*b^2*(a/b)^(2/3) + a* b^4*(a/b)^(2/3))*(a/b)^(1/3)) - 2*(2*a^2*b*(3*(a/b)^(2/3) - 2) - 2*a^3*(a/ b)^(1/3) - a*b^2*(a/b)^(1/3) + b^3)*log(sin(d*x + c)^2 - (a/b)^(1/3)*sin(d *x + c) + (a/b)^(2/3))/(a^5*(a/b)^(2/3) - 2*a^3*b^2*(a/b)^(2/3) + a*b^4*(a /b)^(2/3)) - 4*(a^2*b*(3*(a/b)^(2/3) + 4) + 2*a^3*(a/b)^(1/3) + a*b^2*(a/b )^(1/3) - b^3)*log((a/b)^(1/3) + sin(d*x + c))/(a^5*(a/b)^(2/3) - 2*a^3*b^ 2*(a/b)^(2/3) + a*b^4*(a/b)^(2/3)) + 6*(a*b*sin(d*x + c)^2 - b^2*sin(d*x + c) + a*b)/(a^4 - a^2*b^2 + (a^3*b - a*b^3)*sin(d*x + c)^3) + 9*log(sin(d* x + c) + 1)/(a^2 - 2*a*b + b^2) - 9*log(sin(d*x + c) - 1)/(a^2 + 2*a*b + b ^2))/d
Time = 0.40 (sec) , antiderivative size = 581, normalized size of antiderivative = 0.94 \[ \int \frac {\sec (c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx =\text {Too large to display} \] Input:
integrate(sec(d*x+c)/(a+b*sin(d*x+c)^3)^2,x, algorithm="giac")
Output:
-2/3*a*b*log(abs(b*sin(d*x + c)^3 + a))/(a^4*d - 2*a^2*b^2*d + b^4*d) - 2/ 9*(2*a^8*b^2*d*(-a/b)^(1/3) - 3*a^6*b^4*d*(-a/b)^(1/3) + a^2*b^8*d*(-a/b)^ (1/3) - 4*a^7*b^3*d + 9*a^5*b^5*d - 6*a^3*b^7*d + a*b^9*d)*(-a/b)^(1/3)*lo g(abs(-(-a/b)^(1/3) + sin(d*x + c)))/(a^11*b*d^2 - 4*a^9*b^3*d^2 + 6*a^7*b ^5*d^2 - 4*a^5*b^7*d^2 + a^3*b^9*d^2) - 2/9*((2*sqrt(3)*a^3 + sqrt(3)*a*b^ 2)*(-a*b^2)^(2/3) + (4*sqrt(3)*a^2*b^2 - sqrt(3)*b^4)*(-a*b^2)^(1/3))*arct an(1/3*sqrt(3)*((-a/b)^(1/3) + 2*sin(d*x + c))/(-a/b)^(1/3))/((a^6*b - 2*a ^4*b^3 + a^2*b^5)*d) + 1/9*((2*a^3 + a*b^2)*(-a*b^2)^(2/3) - (4*a^2*b^2 - b^4)*(-a*b^2)^(1/3))*log(sin(d*x + c)^2 + (-a/b)^(1/3)*sin(d*x + c) + (-a/ b)^(2/3))/((a^6*b - 2*a^4*b^3 + a^2*b^5)*d) - 1/2*log(abs(-sin(d*x + c) + 1))/(a^2*d + 2*a*b*d + b^2*d) + 1/2*log(abs(-sin(d*x + c) - 1))/(a^2*d - 2 *a*b*d + b^2*d) + 1/3*(a^3*b - a*b^3 + (a^3*b - a*b^3)*sin(d*x + c)^2 - (a ^2*b^2 - b^4)*sin(d*x + c))/((b*sin(d*x + c)^3 + a)*(a + b)^2*(a - b)^2*a* d)
Time = 36.17 (sec) , antiderivative size = 980, normalized size of antiderivative = 1.59 \[ \int \frac {\sec (c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx =\text {Too large to display} \] Input:
int(1/(cos(c + d*x)*(a + b*sin(c + d*x)^3)^2),x)
Output:
symsum(log(((8*b^6)/27 - (16*a^2*b^4)/27)/(a^7 + a^3*b^4 - 2*a^5*b^2) + ro ot(1458*a^7*b^2*z^3 - 729*a^5*b^4*z^3 - 729*a^9*z^3 - 1458*a^6*b*z^2 - 108 *a^3*b^2*z - 64*a^2*b + 8*b^3, z, k)*(((32*a*b^7)/27 + (128*a^3*b^5)/27)/( a^7 + a^3*b^4 - 2*a^5*b^2) - root(1458*a^7*b^2*z^3 - 729*a^5*b^4*z^3 - 729 *a^9*z^3 - 1458*a^6*b*z^2 - 108*a^3*b^2*z - 64*a^2*b + 8*b^3, z, k)*(root( 1458*a^7*b^2*z^3 - 729*a^5*b^4*z^3 - 729*a^9*z^3 - 1458*a^6*b*z^2 - 108*a^ 3*b^2*z - 64*a^2*b + 8*b^3, z, k)*((16*a^3*b^9 - 77*a^5*b^7 + 34*a^7*b^5 + 27*a^9*b^3)/(a^7 + a^3*b^4 - 2*a^5*b^2) + root(1458*a^7*b^2*z^3 - 729*a^5 *b^4*z^3 - 729*a^9*z^3 - 1458*a^6*b*z^2 - 108*a^3*b^2*z - 64*a^2*b + 8*b^3 , z, k)*((36*a^4*b^10 + 108*a^6*b^8 - 324*a^8*b^6 + 180*a^10*b^4)/(a^7 + a ^3*b^4 - 2*a^5*b^2) + (sin(c + d*x)*(4374*a^5*b^9 - 7290*a^7*b^7 + 1458*a^ 9*b^5 + 1458*a^11*b^3))/(27*(a^7 + a^3*b^4 - 2*a^5*b^2))) + (sin(c + d*x)* (216*a^2*b^10 - 864*a^4*b^8 - 1836*a^6*b^6 + 2484*a^8*b^4))/(27*(a^7 + a^3 *b^4 - 2*a^5*b^2))) + ((64*a^2*b^8)/9 - (353*a^4*b^6)/9 + (388*a^6*b^4)/9) /(a^7 + a^3*b^4 - 2*a^5*b^2) + (sin(c + d*x)*(96*a*b^9 - 408*a^3*b^7 + 447 *a^5*b^5))/(27*(a^7 + a^3*b^4 - 2*a^5*b^2))) + (sin(c + d*x)*(16*b^8 + 134 *a^2*b^6 - 236*a^4*b^4))/(27*(a^7 + a^3*b^4 - 2*a^5*b^2))) + (8*a*b^5*sin( c + d*x))/(9*(a^7 + a^3*b^4 - 2*a^5*b^2)))*root(1458*a^7*b^2*z^3 - 729*a^5 *b^4*z^3 - 729*a^9*z^3 - 1458*a^6*b*z^2 - 108*a^3*b^2*z - 64*a^2*b + 8*b^3 , z, k), k, 1, 3)/d - log(sin(c + d*x) - 1)/(d*(4*a*b + 2*a^2 + 2*b^2))...
\[ \int \frac {\sec (c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {too large to display} \] Input:
int(sec(d*x+c)/(a+b*sin(d*x+c)^3)^2,x)
Output:
(8608*cos(c + d*x)*sin(c + d*x)**2*a**3*b**3 - 1216*cos(c + d*x)*sin(c + d *x)**2*a*b**5 + 1560*cos(c + d*x)*sin(c + d*x)*a**4*b**2 - 2460*cos(c + d* x)*sin(c + d*x)*a**2*b**4 - 2304*cos(c + d*x)*sin(c + d*x)*b**6 - 13120*co s(c + d*x)*a**3*b**3 + 832*cos(c + d*x)*a*b**5 + 194688*int(tan((c + d*x)/ 2)**4/(13*tan((c + d*x)/2)**14*a**4 + 62*tan((c + d*x)/2)**14*a**2*b**2 + 65*tan((c + d*x)/2)**12*a**4 + 310*tan((c + d*x)/2)**12*a**2*b**2 + 208*ta n((c + d*x)/2)**11*a**3*b + 992*tan((c + d*x)/2)**11*a*b**3 + 117*tan((c + d*x)/2)**10*a**4 + 558*tan((c + d*x)/2)**10*a**2*b**2 + 416*tan((c + d*x) /2)**9*a**3*b + 1984*tan((c + d*x)/2)**9*a*b**3 + 65*tan((c + d*x)/2)**8*a **4 + 1142*tan((c + d*x)/2)**8*a**2*b**2 + 3968*tan((c + d*x)/2)**8*b**4 - 65*tan((c + d*x)/2)**6*a**4 - 1142*tan((c + d*x)/2)**6*a**2*b**2 - 3968*t an((c + d*x)/2)**6*b**4 - 416*tan((c + d*x)/2)**5*a**3*b - 1984*tan((c + d *x)/2)**5*a*b**3 - 117*tan((c + d*x)/2)**4*a**4 - 558*tan((c + d*x)/2)**4* a**2*b**2 - 208*tan((c + d*x)/2)**3*a**3*b - 992*tan((c + d*x)/2)**3*a*b** 3 - 65*tan((c + d*x)/2)**2*a**4 - 310*tan((c + d*x)/2)**2*a**2*b**2 - 13*a **4 - 62*a**2*b**2),x)*sin(c + d*x)**3*a**7*b**3*d - 516672*int(tan((c + d *x)/2)**4/(13*tan((c + d*x)/2)**14*a**4 + 62*tan((c + d*x)/2)**14*a**2*b** 2 + 65*tan((c + d*x)/2)**12*a**4 + 310*tan((c + d*x)/2)**12*a**2*b**2 + 20 8*tan((c + d*x)/2)**11*a**3*b + 992*tan((c + d*x)/2)**11*a*b**3 + 117*tan( (c + d*x)/2)**10*a**4 + 558*tan((c + d*x)/2)**10*a**2*b**2 + 416*tan((c...